Electrons kinematical properties of course obey Newton dynamics, but you have to take into account, that even single electron alone is charged particle.
If it is under influence of any force ( and in fact the only possible forces here are fundamental quantum fields: electroweak or gravitation) it get some acceleration, and then radiates electromagnetic fields. That's why single electron in classical regime is not described by Newton kinematics alone ( as for example a thrown stone), but by theory of electrodynamics in classical limit, where number of particles is constant.
You may imagine that radiated electromagnetic field take some energy from electron. That's why electrons are hard to accelerate, because this radiation factor grows very rapidly as function of velocity. Magnetic and electric fields generated by electron move, transforms itself exactly in a way predicted by Maxwell-Clark equations. Even when there is no acceleration, constant velocity only, radiationelectromagnetic field still is present, which is deeply connected to special relativity theory ( and the first paper Einstein wrote about SRT was titled "On the move of electrodynamics of moving bodies"). ( Edited unfortunate words about radiation).